reserve i,j,k for Nat;
reserve K for non empty addLoopStr,
  a for Element of K,
  p for FinSequence of the carrier of K,
  R for Element of i-tuples_on the carrier of K;
reserve K for left_zeroed right_zeroed add-associative right_complementable
  non empty addLoopStr,
  R,R1,R2 for Element of i-tuples_on the carrier of K;
reserve K for non empty addLoopStr,
  a1,a2 for Element of K,
  p1,p2 for FinSequence of the carrier of K,
  R1,R2 for Element of i-tuples_on the carrier of K;
reserve K for Abelian right_zeroed add-associative right_complementable non
  empty addLoopStr,
  R,R1,R2,R3 for Element of i-tuples_on the carrier of K;
reserve K for non empty multMagma,
  a,a9,a1,a2 for Element of K,
  p for FinSequence of the carrier of K,
  R for Element of i-tuples_on the carrier of K;
reserve K for distributive non empty doubleLoopStr,
  a,a1,a2 for Element of K ,
  R,R1,R2 for Element of i-tuples_on the carrier of K;
reserve K for non empty multMagma,
  a1,a2,b1,b2 for Element of K,
  p1,p2 for FinSequence of the carrier of K,
  R1,R2 for Element of i-tuples_on the carrier of K;
reserve K for commutative non empty multMagma,
  p,q for FinSequence of the carrier of K,
  R1,R2 for Element of i-tuples_on the carrier of K;

theorem Th64:
  mlt(p,q)=mlt(q,p)
proof
  reconsider r=mlt(p,q) as FinSequence of the carrier of K;
  reconsider s=mlt(q,p) as FinSequence of the carrier of K;
  reconsider k=min(len p,len q) as Element of NAT by XXREAL_0:15;
A1: len r =min(len p,len q) by FINSEQ_2:71;
  then
A2: dom r = Seg k by FINSEQ_1:def 3;
  min(len p,len q)<=len q by XXREAL_0:17;
  then Seg k c= Seg len q by FINSEQ_1:5;
  then
A3: Seg k c= dom q by FINSEQ_1:def 3;
  min(len p,len q)<= len p by XXREAL_0:17;
  then Seg k c= Seg len p by FINSEQ_1:5;
  then
A4: Seg k c= dom p by FINSEQ_1:def 3;
A5: len s=min(len q,len p) by FINSEQ_2:71;
  then
A6: dom s=Seg k by FINSEQ_1:def 3;
A7: dom r=Seg k by A1,FINSEQ_1:def 3;
  now
    let i be Nat;
    assume
A8: i in dom r;
    then reconsider d1=(p.i),d2=(q.i) as Element of K by A4,A3,A2,FINSEQ_2:11;
    thus r.i=d1*d2 by A8,FUNCOP_1:22
      .= d2*d1
      .=s.i by A7,A6,A8,FUNCOP_1:22;
  end;
  hence thesis by A1,A5,FINSEQ_2:9;
end;
